Explanation Given: The logistic model is, d p d t = ( a − b p ) p ; p ( t 0 ) = p 0 Approach: The equilibrium solution is given by d x d t = 0 . Calculation: The logistic model is given as, d p d t = ( a − b p ) p … ( 1 ) Solve the equation ( 1 ) as, ∫ d p ( a − b p ) p = ∫ d t ∫ [ 1 a p + b a ( a − b p ) ] d p = ∫ d t 1 a ln p − 1 a ln ( a − b p ) = t + C Further solve, 1 a [ ln ( p ( a − b p ) ) ] = t + C ln ( p ( a − b p ) ) = a t + C a p ( a − b p ) = e a t + C a p = a e a t + C a − b p e a t + C a Further solve,

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 3.2, Problem 11E

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Any solution to the given logistic model approaches equilibrium solution p(t)≡a/b as t approaches +∞.

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Chapter 3.2, Problem 10E

Chapter 3.2, Problem 12E

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(c) Describe the distribution plan and show the total distribution cost. Optimal Solution Amount Cost $ 2000 Southern-Hamilton 200 Southern-Butler $ Southern-Clermont 300 4500 Northwest-Hamilton 200 $2400 Northwest-Butler 200 $3000 Northwest-Clermont $ Total Cost ક (d) Recent residential and industrial growth in Butler County has the potential for increasing demand by 100 units. (i) Create an updated distribution plan assuming Southern Gas becomes the preferred supplier. Distribution Plan with Southern Gas Amount Southern-Hamilton $ Cost × Southern-Butler x $ Southern-Clermont 300 $ 4500 Northwest-Hamilton 64 x Northwest-Butler $ × Northwest-Clermont 0 $0 Total Cost $ (ii) Create an updated distribution plan assuming Northwest Gas becomes the preferred supplier. Distribution Plan with Northwest Gas Southern-Hamilton Southern-Butler 0 Southern-Clermont Northwest-Hamilton Northwest-Butler Northwest-Clermont Total Cost Amount × x x +7 $0 Cost × $ × $ × +4 $ -/+ $ × ×

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Fundamentals of Differential Equations and Boundary Value Problems

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Any solution to the given logistic model approaches equilibrium solution p ( t ) ≡ a / b as t approaches + ∞ .

Solution Summary: The author explains that any solution to the given logistic model approaches equilibrium solution p(t)equiv a/b as +infty .

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Any solution to the given logistic model approaches equilibrium solution p(t)≡a/b as t approaches +∞.

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Any solution to the given logistic model approaches equilibrium solution p ( t ) ≡ a / b as t approaches + ∞ .

Solution Summary: The author explains that any solution to the given logistic model approaches equilibrium solution p(t)equiv a/b as +infty .

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To show: Any solution to the given logistic model approaches equilibrium solution p(t)≡a/b as t approaches +∞.

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Chapter 3 Solutions

To show:

Any solution to the given logistic model approaches equilibrium solution p(t)≡a/b as t approaches +∞.